Scaling and parameterizing a controller

ABSTRACT

Controller scaling and parameterization are described. Techniques that can be improved by employing the scaling and parameterization include, but are not limited to, controller design, tuning and optimization. The scaling and parameterization methods described here apply to transfer function based controllers, including PID controllers. The parameterization methods also applies to state feedback and state observer based controllers, as well as linear active disturbance rejection controllers.

CROSS REFERENCE TO RELATED APPLICATIONS

This application claims the benefit of U.S. Provisional Application60/373,404 filed Apr. 18, 2002, which is incorporated herein byreference.

TECHNICAL FIELD

The systems, methods, application programming interfaces (API),graphical user interfaces (GUI), computer readable media, and so ondescribed herein relate generally to controllers and more particularlyto scaling and parameterizing controllers, which facilitates improvingcontroller design, tuning, and optimizing.

BACKGROUND

A feedback (closed-loop) control system 10, as shown in Prior Art FIG.1, is widely used to modify the behavior of a physical process, denotedas the plant 110, so it behaves in a specific desirable way over time.For example, it may be desirable to maintain the speed of a car on ahighway as close as possible to 60 miles per hour in spite of possiblehills or adverse wind; or it may be desirable to have an aircraft followa desired altitude, heading and velocity profile independently of windgusts; or it may be desirable to have the temperature and pressure in areactor vessel in a chemical process plant maintained at desired levels.All these are being accomplished today by using feedback control, andthe above are examples of what automatic control systems are designed todo, without human intervention.

The key component in a feedback control system is the controller 120,which determines the difference between the output of the plant 110,such as measured by sensor 130 (e.g., the temperature) and its desiredvalue and produces a corresponding control signal u (e.g., turning aheater on or off). The goal of controller design is usually to make thisdifference as small as possible as soon as possible. Today, controllersare employed in a large number of industrial control applications and inareas like robotics, aeronautics, astronautics, motors, motion control,thermal control, and so on.

Classic Controllers:

Classic Control Theory provides a number of techniques an engineer canuse in controller design. Existing controllers for linear, timeinvariant, and single-input single output plants can be categorized intothree forms: the proportional/integral/derivative (PID) controllers,transfer function based (TFB) controllers, and state feedback (SF)controllers. The PID controller is defined by the equation

$\begin{matrix}{u = {{K_{p}e} + {K_{I}{\int e}} + {K_{D}\overset{.}{e}}}} & (1)\end{matrix}$where u is the control signal and e is the error between the setpointand the process output being controlled. This type of controller hasbeen employed in engineering and other applications since the early1920s. It is an error based controller that does not require an explicitmathematical model of the plant. The TFB controller is given in the formof

$\begin{matrix}{{{U(s)} = {{G_{c}(s)}{E(s)}}},{{G_{c}(s)} = \frac{n(s)}{d(s)}}} & (2)\end{matrix}$where U(s) and E(s) are Laplace Transforms of u and e defined above, andn(s) and d(s) are polynomials in s. The TFB controller can be designedusing methods in control theory based on the transfer function model ofthe plant, G_(p)(s). A PID controller can be considered a special caseof a TFB controller because it has an equivalent transfer function of

$\begin{matrix}{{G_{c}(s)} = {k_{p} + \frac{k_{i}}{s} + {k_{d}s}}} & (3)\end{matrix}$The State Feedback (SF) Controller

The SF controller can be defined byu=r+K{circumflex over (x)}  (4)and is based on the state space model of the plant:{dot over (x)}(t)=Ax(t)+Bu(t), y(t)=Cx(t)+Du(t)  (5)When the state x is not accessible, a state observer (SO):{dot over ({circumflex over (x)}=A{circumflex over (x)}+Bu+L(y−ŷ)  (6)is often used to find its estimate, {circumflex over (x)}. Here r is thesetpoint for the output to follow.

In addition to the above controllers, a more practical controller is therecently developed Active Disturbance Rejection Controller (ADRC). Itslinear form (LADRC) for a second order plant is introduced below as anillustration. The unique distinction of ADRC is that it is largelyindependent of the mathematical model of the plant and is thereforebetter than most controllers in performance and robustness in practicalapplications.

Linear Activated Disturbance Rejection Controller (LADRC)

Consider an example of controlling a second order plantÿ=−a{dot over (y)}−by+w+bu  (7)where y and u are output and input, respectively, and w is an inputdisturbance. Here both parameters, a and b, are unknown, although thereis some knowledge of b, (e.g., b₀≈b, derived from the initialacceleration of y in step response). Rewrite (7) asÿ=−a{dot over (y)}−by+w+(b−b ₀)u+b ₀ u=f+b ₀ u  (8)where f=−a{dot over (y)}−by+w+(b−b₀)u. Here f is referred to as thegeneralized disturbance, or disturbance, because it represents both theunknown internal dynamics, −a{dot over (y)}−by+(b−b₀)u and the externaldisturbance w(t).

If an estimate of f, {circumflex over (f)} can be obtained, then thecontrol law

$u = \frac{{- \hat{f}} + u_{0}}{b_{0}}$reduces the plant to ÿ=(f−{circumflex over (f)})+u₀ which is a unit-gaindouble integrator control problem with a disturbance (f−{circumflex over(f)}).

Thus, rewrite the plant in (8) in state space form as

$\begin{matrix}\left\{ \begin{matrix}{{\overset{.}{x}}_{1} = x_{2}} \\{{\overset{.}{x}}_{2} = {x_{3} + {b_{0}u}}} \\{{\overset{.}{x}}_{3} = h} \\{y = x_{1}}\end{matrix} \right. & (9)\end{matrix}$with x₃=f added as an augmented state, and h={dot over (f)} is seen asan unknown disturbance. Now f can be estimated using a state observerbased on the state space model

$\begin{matrix}{{\overset{.}{x} = {{Ax} + {Bu} + {Eh}}}{y = {Cz}}{where}{{A = \begin{bmatrix}0 & 1 & 0 \\0 & 0 & 1 \\0 & 0 & 0\end{bmatrix}},{B = {{\begin{bmatrix}0 \\b_{0} \\0\end{bmatrix}.C} = \left\lbrack {1\mspace{14mu} 0\mspace{14mu} 0} \right\rbrack}},{E = \begin{bmatrix}0 \\0 \\1\end{bmatrix}}}} & (10)\end{matrix}$Now the state space observer, denoted as the linear extended stateobserver (LESO), of (10) can be constructed asż=Az+Bu+L(y−ŷ)ŷ=Cz  (11)

which can be reconstructed in software, for example, and L is theobserver gain vector, which can be obtained using various methods knownin the art like pole placement,L=[β₁β₂β₃]^(T)  (12)where [ ]^(T) denotes transpose. With the given state observer, thecontrol law can be given as:

$\begin{matrix}{u = \frac{{- z_{3}} + u_{0}}{b_{0}}} & (13)\end{matrix}$Ignoring the inaccuracy of the observer,ÿ=(f−z ₃)+u ₀ ≈u ₀  (14)which is an unit gain double integrator that can be implemented with aPD controlleru ₀ =k _(p)(r−z ₁)−k _(d) z ₂  (15)Controller Tuning

Over the years, the advances in control theory provided a number ofuseful analysis and design tools. As a result, controller design movedfrom empirical methods (e.g., ad hoc tuning via Ziegler and Nicholstuning tables for PID) to analytical methods (e.g., pole placement). Thefrequency response method (Bode and Nyquist plots) also facilitatedanalytical control design.

Conventionally, controllers are individually designed according todesign criteria and then individually tuned until they exhibit anacceptable performance. Practicing engineers may design controllers,(e.g., PID) using look-up tables and then tune the controllers usingtrial and error techniques. But each controller is typicallyindividually designed, tuned, and tested.

Tuning controllers has perplexed engineers. Controllers that aredeveloped based on a mathematical model of the plant usually need theirparameters to be adjusted, or “tuned” as they are implemented inhardware and tested. This is because the mathematical model often doesnot accurately reflect the dynamics of the plant. Determiningappropriate control parameters under such circumstances is oftenproblematic, leading to control solutions that are functional butill-tuned, yielding lost performance and wasted control energy.Additionally, and/or alternatively, engineers design using analytical(e.g., pole placement) techniques, but once again tune with trial anderror techniques. Since many industrial machines and engineeringapplications are built to be inherently stable, acceptable controllerscan be designed and tuned using these conventional techniques, however,acceptable performance may not approach optimal performance.

One example conventional technique for designing a PID controllerincluded obtaining an open-loop response and determining what, ifanything, needed to be improved. By way of illustration, the designerwould build a candidate system with a feedback loop, guess the initialvalues of the three gains (e.g., k_(p), k_(d), k_(i)) in PID and observethe performance in terms of rise time, steady state error and so on.Then, the designer might modify the proportional gain to improve risetime. Similarly, the designer might add or modify a derivativecontroller to improve overshoot and an integral controller to eliminatesteady state error. Each component would have its own gain that would beindividually tuned. Thus, conventional designers often faced choosingthree components in a PID controller and individually tuning eachcomponent. Furthermore, there could be many more parameters that thedesign engineer must tune if a TFB or a state feedback state observer(SFSOB) controller is employed.

Another observation of control design is that it is not portable. Thatis, each control problem is solved individually and its solution cannotbe easily modified for another control problem. This means that thetedious design and tuning process must be repeated for each controlproblem.

Thus, having reviewed controllers, the application now describes examplesystems and methods related to controllers.

SUMMARY

This section presents a simplified summary of methods, systems, computerreadable media and so on for scaling and parameterizing controllers tofacilitate providing a basic understanding of these items. This summaryis not an extensive overview and is not intended to identify key orcritical elements of the methods, systems, computer readable media, andso on or to delineate the scope of these items. This summary provides aconceptual introduction in a simplified form as a prelude to the moredetailed description that is presented later.

The application describes scaling and parameterizing controllers. Withthese two techniques, controller designing, tuning, and optimizing canbe improved. In one example, systems, methods, and so on describedherein facilitate reusing a controller design by scaling a controllerfrom one application to another. This scaling may be available, forexample, for applications whose plant differences can be detailedthrough frequency scale and/or gain scale. While PID controllers areused as examples, it is to be appreciated that other controllers canbenefit from scaling and parameterizing as described herein.

Those familiar with filter design understand that filters may bedesigned and then scaled for use in analogous applications. Filterdesigners are versed in the concept of the unit filter which facilitatesscaling filters. In example controller scaling techniques, a planttransfer function is first reduced to a unit gain and unit bandwidth(UGUB) form. Then, a known controller for an appropriate UGUB plant isscaled for an analogous plant. Since certain plants share certaincharacteristics, classes of UGUB plants can be designed for whichcorresponding classes of scaleable, parameterizable controllers can bedesigned.

Since certain classes of plants have similar properties, it is possibleto frequency scale controllers within classes. For example, an anti-lockbrake plant for a passenger car that weighs 2000 pounds may share anumber of characteristics with an anti-lock brake plant for a passengercar that weighs 2500 pounds. Thus, if a UGUB plant can be designed forthis class of cars, then a frequency scaleable controller can also bedesigned for the class of plants. Then, once a controller has beenselected and engineered for a member of the class (e.g., the 2000 poundcar), it becomes a known controller from which other analogouscontrollers can be designed for other similar cars (e.g., the 2500 poundcar) using frequency scaling.

This scaling method makes a controller “portable”. That is a singlecontroller can be used as the “seed” to generate controllers for a largenumber of different plants that are similar in nature. The remainingquestion concerns how to account for differences in design requirements.Controller parameterization addresses this issue. The exampleparameterization techniques described herein make controllercoefficients functions of a single design parameter, namely thecrossover frequency (also known as the bandwidth). In doing so, thecontroller can be tuned for different design requirements, which isprimarily reflected in the bandwidth requirement.

The combination of scaling and parameterization methods means that anexisting controller (including PID, TFB, and SFSOB) can be scaled fordifferent plants and then, through the adjustment of one parameter,changed to meet different performance requirements that are unique indifferent applications.

Certain illustrative example methods, systems, computer readable mediaand so on are described herein in connection with the followingdescription and the annexed drawings. These examples are indicative,however, of but a few of the various ways in which the principles of themethods, systems, computer readable media and so on may be employed andthus are intended to be inclusive of equivalents. Other advantages andnovel features may become apparent from the following detaileddescription when considered in conjunction with the drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

Prior Art FIG. 1 illustrates the configuration of an output feedbackcontrol system.

FIG. 2 illustrates a feedback control configuration.

FIG. 3 illustrates an example controller production system.

FIG. 4 illustrates an example controller scaling method.

FIG. 5 illustrates an example controller scaling method.

FIG. 6 compares controller responses.

FIG. 7 illustrates loop shaping.

FIG. 8 illustrates a closed loop simulator setup.

FIG. 9 compares step responses.

FIG. 10 illustrates transient profile effects.

FIG. 11 compares PD and LADRC controllers.

FIG. 12 illustrates LESO performance.

FIG. 13 is a flowchart of an example design method.

FIG. 14 is a schematic block diagram of an example computingenvironment.

FIG. 15 illustrates a data packet.

FIG. 16 illustrates sub-fields within a data packet.

FIG. 17 illustrates an API.

FIG. 18 illustrates an example observer based system.

Lexicon

As used in this application, the term “computer component” refers to acomputer-related entity, either hardware, firmware, software, acombination thereof, or software in execution. For example, a computercomponent can be, but is not limited to being, a process running on aprocessor, a processor, an object, an executable, a thread of execution,a program and a computer. By way of illustration, both an applicationrunning on a server and the server can be computer components. One ormore computer components can reside within a process and/or thread ofexecution and a computer component can be localized on one computerand/or distributed between two or more computers.

“Computer communications”, as used herein, refers to a communicationbetween two or more computers and can be, for example, a networktransfer, a file transfer, an applet transfer, an email, a hypertexttransfer protocol (HTTP) message, a datagram, an object transfer, abinary large object (BLOB) transfer, and so on. A computer communicationcan occur across, for example, a wireless system (e.g., IEEE 802.11), anEthernet system (e.g., IEEE 802.3), a token ring system (e.g., IEEE802.5), a local area network (LAN), a wide area network (WAN), apoint-to-point system, a circuit switching system, a packet switchingsystem, and so on.

“Logic”, as used herein, includes but is not limited to hardware,firmware, software and/or combinations of each to perform a function(s)or an action(s). For example, based on a desired application or needs,logic may include a software controlled microprocessor, discrete logicsuch as an application specific integrated circuit (ASIC), or otherprogrammed logic device. Logic may also be fully embodied as software.

An “operable connection” is one in which signals and/or actualcommunication flow and/or logical communication flow may be sent and/orreceived. Usually, an operable connection includes a physical interface,an electrical interface, and/or a data interface, but it is to be notedthat an operable connection may consist of differing combinations ofthese or other types of connections sufficient to allow operablecontrol.

“Signal”, as used herein, includes but is not limited to one or moreelectrical or optical signals, analog or digital, one or more computerinstructions, a bit or bit stream, or the like.

“Software”, as used herein, includes but is not limited to, one or morecomputer readable and/or executable instructions that cause a computeror other electronic device to perform functions, actions and/or behavein a desired manner. The instructions may be embodied in various formslike routines, algorithms, modules, methods, threads, and/or programs.Software may also be implemented in a variety of executable and/orloadable forms including, but not limited to, a stand-alone program, afunction call (local and/or remote), a servelet, an applet, instructionsstored in a memory, part of an operating system or browser, and thelike. It is to be appreciated that the computer readable and/orexecutable instructions can be located in one computer component and/ordistributed between two or more communicating, co-operating, and/orparallel processing computer components and thus can be loaded and/orexecuted in serial, parallel, massively parallel and other manners. Itwill be appreciated by-one of ordinary skill in the art that the form ofsoftware may be dependent on, for example, requirements of a desiredapplication, the environment in which it runs, and/or the desires of adesigner/programmer or the like.

“Data store”, as used herein, refers to a physical and/or logical entitythat can store data. A data store may be, for example, a database, atable, a file, a list, a queue, a heap, and so on. A data store mayreside in one logical and/or physical entity and/or may be distributedbetween two or more logical and/or physical entities.

To the extent that the term “includes” is employed in the detaileddescription or the claims, it is intended to be inclusive in a mannersimilar to the term “comprising” as that term is interpreted whenemployed as a transitional word in a claim.

To the extent that the term “or” is employed in the claims (e.g., A orB) it is intended to mean “A or B or both”. When the author intends toindicate “only A or B but not both”, then the author will employ theterm “A or B but not both”. Thus, use of the term “or” in the claims isthe inclusive, and not the exclusive, use. See BRYAN A. GARNER, ADICTIONARY OF MODERN LEGAL USAGE 624 (2d Ed. 1995).

DETAILED DESCRIPTION

Example methods, systems, computer media, and so on are now describedwith reference to the drawings, where like reference numerals are usedto refer to like elements throughout. In the following description forpurposes of explanation, numerous specific details are set forth inorder to facilitate thoroughly understanding the methods, systems,computer readable media, and so on. It may be evident, however, that themethods, systems and so on can be practiced without these specificdetails. In other instances, well-known structures and devices are shownin block diagram form in order to simplify description.

Scaling:

Controllers typically are not scalable and thus are not portable betweenapplications. However, controllers can be made portable via scaling asdescribed in the example systems and methods provided herein. Ingeneral, a plant mathematically represented by a transfer functionG_(p)(s), (where s is the Laplace Transform variable) can be scaledaccording to:G _(p)(s)=kG _(p)(s/ω _(p))  (16)where ω_(p) is the plant frequency scale and k is the gain scale, torepresent a large number of plants that differ from the original plantby a frequency scale, ω_(p), and a gain scale, k.

Then, a corresponding controller G_(c)(s) for the plant G_(p)(s) can bescaled according to:G _(c)(s)=(1/k)G _(c)(s/ω _(p)).  (17)

Consider a unit feedback control system 200 with the plant G_(p)(s) 210and controller G_(c)(s) 220, as shown in FIG. 2. Assume that G_(c)(s)220 was designed with desired command following, disturbance rejection,noise rejection, and stability robustness. Now, consider a similar classof plants kG_(p)(s/ω_(p)). For given ω_(p), using example systems andmethods described herein, a suitable controller can be produced throughfrequency scaling. Thus define ω_(p) as the frequency scale and k as thegain scale of the plant G_(p)(s/ω_(p)) with respect toG_(p)(s).  (18)ThenG _(c)(s)=(1/k)G _(c)(s/ω _(p)).  (19)

Referring to FIG. 3, an example system 300 that employs frequencyscaling is illustrated. The system 300 includes a controller identifier310 that can identify a known controller associated with controlling aknown plant. The controller may have one or more scaleable parameters(e.g., frequency, gains) that facilitate scaling the controller. Thecontroller identifier 310 may access a controller information data store330 and/or a plant information data store 340 to facilitatecharacterizing one or more properties of the known controller. By way ofillustration, the controller identifier 310 may identify the frequencyscale of the controller (ω_(c)) and/or the frequency scale (ω_(p)) andtransfer function (s) of a plant controlled by the known controller.

The controller information data store 330 may store, for example,controller class information and/or information concerning scaleablecontroller parameters. Similarly, the plant data store 340 may store,for example, plant information like transfer function shape, frequencyscale, and so on.

The system 300 may also include a controller scaler 320 that produces ascaled controller from the identified scaleable parameter. The scaler320 may make scaling decisions based, for example, on information in thecontroller information data store 330 (e.g., controller class, scaleableparameters, frequency scale), information in the plant information datastore 340 (e.g. plant class, plant transfer function, frequency scale),and so on.

While illustrated as two separate entities, it is to be appreciated thatthe identifier 310 and scaler 320 could be implemented in a singlecomputer component and/or as two or more distributed, communicating,co-operating computer components. Thus, the entities illustrated in FIG.3 may communicate through computer communications using signals, carrierwaves, data packets, and so on. Similarly, while illustrated as twoseparate data stores, the controller information data store 330 and theplant information data store 340 may be implemented as a single datastore and/or distributed between two or more communicating, co-operatingdata stores.

Aspects of controller scaling can be related to filter design. In filterdesign, with the bandwidth, the pass band, and stop band requirementsgiven, filter design is straight forward. An example filter designmethod includes finding a unit bandwidth filter, such as an nth orderChebeshev filter H(s), that meets the pass band and stop bandspecifications and then frequency scaling the filter as H(s/ω₀) toachieve a bandwidth of ω₀.

Revisiting the system 200 in FIG. 2, to facilitate understandingfrequency scaling and time scaling as related to controllers, denoteω_(p) as the frequency scale of the plant G_(p)(s/ω_(p)) with respect toG_(p)(s) 210, and τ_(p)=1/ω_(p), the corresponding time scale. Thendenote k as the gain scale of the plant kG_(p)(s) with respect toG_(p)(s) 210. With these definitions in hand, differences in exampleindustrial control problems can be described in terms of the frequencyand gain scales. For example, temperature processes with different timeconstants (in first order transfer functions), motion control problemswith different inertias, motor sizes, frictions, and the like can bedescribed in terms of the defined frequency and gain scales.

These scales facilitate paying less attention to differences betweencontrollers and applications and more attention to a generic solutionfor a class of problems because using the scales facilitates reducinglinear time invariant plants, proper and without a finite zero, to oneof the following example forms:

$\begin{matrix}{\frac{1}{s + 1},\frac{1}{s},\frac{1}{s^{2} + {2\xi\; s} + 1},\frac{1}{s\left( {s + 1} \right)},\frac{1}{s^{2}},\frac{1}{s^{3} + {\xi_{1}s^{2}} + {\xi_{2}s} + 1},} & (22)\end{matrix}$through gain and frequency scaling. For example, the motion controlplant of G_(p)(s)=23.2/s(s+1.41) is a variation of a generic motioncontrol plant G_(p)(s)=1/s(s+1) with a gain factor of k=11.67 andω_(p)=1.41.

$\begin{matrix}{\frac{23.2}{s\left( {s + 1.41} \right)} = \frac{11.67}{\frac{s}{1.41}\left( {\frac{s}{1.41} + 1} \right)}} & (23)\end{matrix}$

Equation (22) describes many example industrial control problems thatcan be approximated by a first order or a second order transfer functionresponse. Additionally, equation (22) can be appended by terms like:

$\begin{matrix}{\frac{s + 1}{s^{2} + {2\xi\; s} + 1},\frac{s^{2} = {{2\xi_{z}s} + 1}}{s^{3} + {\xi_{1}s^{2}} + {\xi_{2}s} + 1},} & (24)\end{matrix}$to include systems with finite zeros. Thus, while a set of examples isprovided in equations (22) and (24), it is to be appreciated that agreater and/or lesser number of forms can be employed in accordance withthe systems and methods described herein. Furthermore, in some examples,scaling can be applied to reflect the unique characteristics of certainproblems. For example, a motion control system with significant resonantproblems can be modeled and scaled as

$\begin{matrix}{{\frac{k}{\frac{s}{\omega_{p}}\left( {\frac{s}{\omega_{p}} + 1} \right)}\frac{\left( \frac{s}{\omega_{rz}} \right)^{2} + {2\xi_{z}\frac{s}{\omega_{rz}}} + 1}{\left( \frac{s}{\omega_{rp}} \right)^{2} + {2\xi_{p}\frac{s}{\omega_{rp}}} + 1}}\mspace{149mu}\left. {scaling}\mspace{14mu}\Downarrow \right.} & \; \\{\frac{1}{s\left( {s + 1} \right)}\frac{\left( \frac{s}{m} \right)^{2} + {2\xi_{z}\frac{s}{m}} + 1}{\left( \frac{s}{n} \right)^{2} + {2\xi_{p}\frac{s}{n}} + 1}} & (25)\end{matrix}$where the resonant frequencies satisfy ω_(rp)=nω_(p), ω_(rz)=mω_(p).Problems with multiple frequency scales, ω_(p), nω_(p), and mω_(p), canbe referred to as multi-scale problems. With these definitions in hand,an example controller scaling technique is now described.

Assume G_(c)(s) is a stabilizing controller for plant G_(p)(s), and theloop gain crossover frequency is ω_(c), then the controllerG _(c)(s)=G _(c)(s/ω _(p))/k  (26)will stabilize the plant G _(p)(s)=kG_(p1)(s/ω_(p)). The new controllernew loop gainL (s)= G _(p)(s) G _(c)(s)  (27)will have a bandwidth of ω_(c)ω_(p), and substantially the samestability margins ofL(s)=G _(p)(s)G _(c)(s)sinceL (s)=L(s/ω _(p)).Note that the new closed-loop system has substantially the samefrequency response shape as the original system except that it isshifted by ω_(p). Thus, feedback control properties like bandwidth,disturbance and noise rejection are retained, as is the stabilityrobustness, from the previous design, except that frequency ranges areshifted by ω_(p).

Now that controller scaling has been described, PID scaling can beaddressed. According to the frequency scale principle discussed above,and assuming the original controller for G_(p)(s) is a PID, e.g.,

$\begin{matrix}{{G_{c}(s)} = {k_{p} + \frac{k_{i}}{s} + {k_{d}s}}} & (28)\end{matrix}$then the new controller for the plant kG_(p)(s/ω_(p)) is obtained from(28) as

$\begin{matrix}{{G_{c}(s)} = {\left( {k_{p} + {k_{i}\frac{\omega_{p}}{s}} + {k_{d}\frac{s}{\omega_{p}}}} \right)/k}} & (29)\end{matrix}$That is, the new PID gains, k _(p), k _(i), and k _(d) are obtained fromthe original ones as

$\begin{matrix}{{{\overset{\_}{k}}_{p} = \frac{k_{p}}{k}},{{\overset{\_}{k}}_{i} = \frac{k_{i}\omega_{p}}{k}},{{\overset{\_}{k}}_{d} = \frac{k_{d}}{k\;\omega_{p}}}} & (30)\end{matrix}$

To demonstrate the practical application and tangible results possiblefrom the method described above, in the following example, consider aplant that has a transfer function of

${G_{p}(s)} = \frac{1}{s^{2} + s + 1}$and the PID control gains of k_(p)=3, k_(i)=1, and k_(d)=2. Now, assumethe plant has changed to

${G_{p}(s)} = \frac{1}{\left( \frac{s}{10} \right)^{2} + \frac{s}{10} + 1}$The new gains are calculated from equation (30) as k _(p)=3, k _(i)=10,k _(d)=0.2. Thus, rather than having to build, design, and tune thecontroller for the plant

${G_{p}(s)} = \frac{1}{\left( \frac{s}{10} \right)^{2} + \frac{s}{10} + 1}$from scratch, the PID designer was able to select an existing PIDappropriate for the PID class and scale the PID. Thus, frequency scalingfacilitates new systems and methods for controller design that takeadvantage of previously designed controllers and the relationshipsbetween controllers in related applications.

In one example, the controller is a PID controller. The PID controllermay have a plant frequency scale ω_(p) as a scaleable parameter. Inanother example, the method includes producing the scaled controller.For example, a computer component may be programmed to perform thefrequency scaled controlling. Additionally, computer executable portionsof the method may be stored on a computer readable medium and/or betransmitted between computer components by, for example, carrier wavesencoding computer executable instructions.

In view of the exemplary systems shown and described below, examplemethodologies that are implemented will be better appreciated withreference to the flow diagrams of FIGS. 4, 5 and 13. While for purposesof simplicity of explanation, the illustrated methodologies are shownand described as a series of blocks, it is to be appreciated that themethodologies are not limited by the order of the blocks, as some blockscan occur in different orders and/or concurrently with other blocks fromthat shown and described. Moreover, less than all the illustrated blocksmay be required to implement an example methodology. Furthermore,additional and/or alternative methodologies can employ additional, notillustrated blocks. In one example, methodologies are implemented ascomputer executable instructions and/or operations, stored on computerreadable media including, but not limited to an application specificintegrated circuit (ASIC), a compact disc (CD), a digital versatile disk(DVD), a random access memory (RAM), a read only memory (ROM), aprogrammable read only memory (PROM), an electronically erasableprogrammable read only memory (EEPROM), a disk, a carrier wave, and amemory stick.

In the flow diagrams, rectangular blocks denote “processing blocks” thatmay be implemented, for example, in software. Similarly, the diamondshaped blocks denote “decision blocks” or “flow control blocks” that mayalso be implemented, for example, in software. Alternatively, and/oradditionally, the processing and decision blocks can be implemented infunctionally equivalent circuits like a digital signal processor (DSP),an ASIC, and the like.

A flow diagram does not depict syntax for any particular programminglanguage, methodology, or style (e.g., procedural, object-oriented).Rather, a flow diagram illustrates functional information one skilled inthe art may employ to program software, design circuits, and so on. Itis to be appreciated that in some examples, program elements liketemporary variables, initialization of loops and variables, routineloops, and so on are not shown.

Turning to FIG. 5, a flowchart for an example method 500 for producing acontroller is illustrated. The method 500 includes, at 510, identifyinga controller G_(c)(s) that stabilizes a plant G_(p)(s) where thecontroller has a frequency ω_(c) and, at 520, producing a controller G_(c)(s) by scaling the controller G_(p)(s) according to G_(c)(s)=G_(c)(s/ω_(p))/k, where the controller G _(c)(s) will stabilizethe plant G _(p)(s)=kG_(p1)(s/ω_(p)), where ω_(p) is the frequency scaleof the plant G_(p)(s/ω_(p)), and where k is the gain scale of the plantkG_(p)(s). In one example, the controller is a PID controller of theform

${{G_{c}(s)} = {k_{p} + \frac{k_{i}}{s} + {k_{d}s}}},$where k_(p) is a proportional gain, k_(i) is an integral gain, and k_(d)is a derivative gain. In another example,

${G_{c}(s)} = {\left( {k_{p} + {k_{i}\frac{\omega_{p}}{s}} + {k_{d}\frac{s}{\omega_{p}}}} \right)/{k.}}$In yet another example, the PID gains k _(p), k _(i), and k _(d) areobtained from the k_(p), k_(i) and k_(d) according to

${{\overset{\_}{k}}_{p} = \frac{k_{p}}{k}},{{\overset{\_}{k}}_{i} = \frac{k_{i}\omega_{p}}{k}},{{\overset{\_}{k}}_{d} = {\frac{k_{d}}{k\;\omega_{p}}.}}$It is to be appreciated that this example method can be employed withlinear and/or non-linear PIDs.

Applying a unit step function as the set point, the responses of anoriginal controller and a scaled controller are shown in FIG. 6,demonstrating that the response of the scaled controller issubstantially the same as the response of the original controller, butscaled by τ=1/ω₀. The gain margins of both systems are substantiallyinfinite and the phase margins are both approximately 82.372 degrees.The 0 dB crossover frequency for both systems are 2.3935 and 23.935 r/s,respectively. Thus, the PID scaled by the example method is demonstrablyappropriate for the application.

While the method described above concerned linear PIDs, it is to beappreciated that the method can also be applied to scaling nonlinearPIDs. For example, PID performance can be improved by using nonlineargains in place of the linear ones. For example,u=k _(p) g _(p)(e)+k _(i) ∫g _(i)(e)dt+k _(d) g _(d)(ė)  (31)where g_(p)(e), g_(i)(e), and g_(d)(e) are nonlinear functions. Thenon-linear PIDs can be denoted NPID. Nonlinearities are selected so thatthe proportional control is more sensitive to small errors, the integralcontrol is limited to the small error region—which leads to significantreduction in the associate phase lag—and the differential control islimited to a large error region, which reduces its sensitivity to thepoor signal to noise ratio when the response reaches steady state andthe error is small.

The NPID retains the simplicity of PID and the intuitive tuning. Thesame gain scaling formula (30) will also apply to the NPID controllerwhen the plant changes from G_(p)(s) to kG_(p)(s/ω_(p)).

Scaling facilitates concentrating on normalized control problems likethose defined in (22). This facilitates selecting an appropriatecontroller for an individual problem by using the scaling formula in(26) and the related systems and methods that produce tangible, results(e.g., scaled controller). This further facilitates focusing on thefundamentals of control, like basic assumptions, requirements, andlimitations. Thus, the example systems, methods, and so on describedherein concerning scaling and parameterization can be employed tofacilitate optimizing individual solutions given the physicalconstraints of a problem.

Parameterization

Working with controllers can be simplified if they can be described interms of a smaller set of parameters than is conventionally possible.Typically, a controller (and possibly an observer) may have many (e.g.15) parameters. The systems and methods described herein concerningparameterization facilitate describing a controller in terms of a singleparameter. In one example, controller parameterization concerns makingcontroller parameters functions of a single variable, the controllerbandwidth ω_(c).

Considering the normalized plants in (22) and assuming desiredclosed-loop transfer functions are:

$\begin{matrix}{{\frac{\omega_{c}}{s + \omega_{c}},\frac{\omega_{c}^{2}}{\left( {s + \omega_{c}} \right)^{2}},\frac{\omega_{c}^{3}}{\left( {s + \omega_{c}} \right)^{3}},}\;} & (32)\end{matrix}$then for second order plants, the damping ratio can be set to unity,resulting in two repeated poles at −ω_(c). The same technique can alsobe applied to higher order plants.

Applying pole-placement design to the first and second order plants in(22), a set of example ω_(c) parameterized controllers are obtained andshown in Table I. Information concerning the plants and the relatedcontrollers can be stored, for example, in a data store.

TABLE I Examples of ω_(c)-parameterized controllers G_(p)(s)$\frac{1}{s + 1}$ $\frac{1}{s}$ $\frac{1}{s^{2} + {2{\xi s}} + 1}$$\frac{1}{s\left( {s + 1} \right)}$ $\frac{1}{s^{2}}$ G_(c)(s, ω_(c))$\frac{\omega_{c}\left( {s + 1} \right)}{s}$ ω_(c)$\omega_{c}^{2}\frac{s^{2} + {2{\xi s}} + 1}{s\left( {s + {2\omega_{c}}} \right)}$$\frac{\omega_{c}^{2}\left( {s + 1} \right)}{s + {2\omega_{c}}}$$\frac{\omega_{c}^{2}s}{s + {2\omega_{c}}}$

Loop shaping design can also be parameterized. Loop-shaping refers tomanipulating the loop gain frequency response, L(jω)=G_(p)(jω)G_(c)(jω),as a control design tool. One example loop-shaping method includesconverting design specifications to loop gain constraints, as shown inFIG. 7 and finding a controller G_(c)(jω) to meet the specifications.

As an example of loop shaping, considering the plants of the formG_(p)(s), in Table I, the desired loop gain can be characterized as

$\begin{matrix}{{L(s)} = {{{G_{p}(s)}{G_{c}(s)}} = {\left( \frac{s + \omega_{1}}{s} \right)^{m}\frac{1}{\frac{s}{\omega_{c}} + 1}\frac{1}{\left( {\frac{s}{\omega_{2}} + 1} \right)^{n}}}}} & (33)\end{matrix}$where ω_(c) is the bandwidth, andω₁<ω_(c), ω₂>ω_(c), m≧0, and n≧0  (34)are selected to meet constrains shown in FIG. 7. In the example, both mand n are integers. In one example, default values for ω₁ and ω₂ areω₁=ω_(c)/10 and ω₂=10ω_(c)  (35)which yield a phase margin greater than forty-five degrees.

Once appropriate loop gain constraints are derived and the correspondinglowest order L(s) in (33) is selected, the controller can be determinedfrom

$\begin{matrix}{{{G_{c}(s)} = {\left( \frac{s + \omega_{1}}{s} \right)^{m}\frac{1}{\frac{s}{\omega_{c}} + 1}\frac{1}{\left( {\frac{s}{\omega_{2}} + 1} \right)^{n}}{G_{p}^{- 1}(s)}}}\mspace{14mu}} & (36)\end{matrix}$An additional constraint on n is that

$\begin{matrix}{\frac{1}{\frac{s}{\omega_{c}} + 1}\frac{1}{\left( {\frac{s}{\omega_{2}} + 1} \right)^{n}}{G_{p}^{- 1}(s)}\mspace{14mu}{is}\mspace{14mu}{{proper}.}} & (37)\end{matrix}$This design is valid for plants with a minimum phase. For a non-minimumphase plant, a minimum phase approximation of G_(p) ⁻¹(s) can beemployed.

A compromise between ω₁ and the phase margin can be made by adjusting ω₁upwards, which will improve the low frequency properties at the cost ofreducing phase margin. A similar compromise can be made between phasemargin and ω₂.

Turning to FIG. 4, an example method 400 for scaling a controller isillustrated. The method 400 includes, at 410, identifying a knowncontroller in a controller class where the known controller controls afirst plant. The method 400 also includes, at 420, identifying ascaleable parameter for the known controller. At 430, the method 400includes identifying a desired controller in the controller class, wherethe desired controller controls a second, frequency related plant and at440, establishing the frequency relation between the known controllerand the desired controller. At 450, the method 400 scales the knowncontroller to the desired controller by scaling the scaleable parameterbased, at least in part, on the relation between the known controllerand the desired controller.

Practical Optimization Based on a Hybrid Scaling and ParameterizationMethod

Practical controller optimization concerns obtaining optimal performanceout of existing hardware and software given physical constraints.Practical controller optimization is measured by performancemeasurements including, but not limited to, command following quickness(a.k.a. settling time), accuracy (transient and steady state errors),and disturbance rejection ability (e.g., attenuation magnitude andfrequency range). Example physical constraints include, but are notlimited to, sampling and loop update rate, sensor noise, plant dynamicuncertainties, saturation limit, and actuation signal smoothnessrequirements.

Conventional tuning relies, for example, on minimizing a cost functionlike H₂ and H₂₈. However, conventional cost functions may notcomprehensively reflect the realities of control engineering, and may,therefore, lead to suboptimal tuning. For example, one common costfunction is mathematically attractive but can lead to suboptimalcontroller tuning. Thus, optimizing other criteria, like ω_(c) areconsidered.

A typical industrial control application involves a stable single-inputsingle-output (SISO) plant, where the output represents a measurableprocess variable to be regulated and the input represents the controlactuation that has a certain dynamic relationship to the output. Thisrelationship is usually nonlinear and unknown, although a linearapproximation can be obtained at an operating point via the plantresponse to a particular input excitation, like a step change.

Evaluating performance measurements in light of physical limitationsyields the fact that they benefit from maximum controller bandwidthω_(c). If poles are placed in the same location, then ω_(c) can becomethe single item to tune. Thus, practical PID optimization can beachieved with single parameter tuning. For example, in manufacturing, adesign objective for an assembly line may be to make it run as fast aspossible while minimizing the down time for maintenance and troubleshooting. Similarly, in servo design for a computer hard disk drive, adesign objective may be to make the read/write head position follow thesetpoint as fast as possible while maintaining extremely high accuracy.In automobile anti-lock brake control design, a design objective may beto have the wheel speed follow a desired speed as closely as possible toachieve minimum braking distance.

In the three examples, the design goal can be translated to maximizingcontroller bandwidth ω_(c). There are other industrial control examplesthat lead to the same conclusion. Thus, ω_(c) maximization appears to bea useful criterion for practical optimality. Furthermore, unlike purelymathematical optimization techniques, ω_(c) optimization has real worldapplicability because it is limited by physical constraints. Forexample, sending ω_(c) to infinity may be impractical because it maycause a resulting signal to vary unacceptably.

As an example of how physical limitations may affect ω_(c) optimization,consider digital control apparatus that have a maximum sampling rate anda maximum loop update rate. The maximum sampling rate is a hardwarelimit associated with the Analog to Digital Converter (ADC) and themaximum loop update rate is software limit related to central processingunit (CPU) speed and the control algorithm complexity. Typically,computation speeds outpace sampling rates and therefore only thesampling rate limitation is considered.

As another example, measurement noise may also be considered whenexamining the physical limitations of ω_(c) optimization. For example,the ω_(c) is limited to the frequency range where the accuratemeasurement of the process variable can be obtained. Outside of thisrange, the noise can be filtered using either analog or digital filters.

Plant dynamic uncertainty may also be considered when examining thephysical limitations of ω_(c) optimization. Conventional control designis based on a mathematical description of the plant, which may only bereliable in a low frequency range. Some physical plants exhibit erraticphase distortions and nonlinear behaviors at a relative high frequencyrange. The controller bandwidth is therefore limited to the lowfrequency range where the plant is well behaved and predictable. Tosafeguard the system from instability, the loop gain is reduced wherethe plant is uncertain. Thus, maximizing the bandwidth safely amounts toexpanding the effective (high gain) control to the edge of frequencyrange where the behavior of the plant is well known.

Similarly, actuator saturation and smoothness may also affect design.Although using transient profile helps to decouple bandwidth design andthe transient requirement, limitations in the actuator like saturation,nonlinearities like backlash and hysteresis, limits on rate of change,smoothness requirements based on wear and tear considerations, and so onmay affect the design. For example, in a motion control application witha significant backlash problem in the gearbox, excessively highbandwidth will result in a chattering gearbox and, very likely,premature breakdown. Thus, ω_(c) optimization, because it considersphysical limitations like sampling rate, loop update rate, plantuncertainty, actuator saturation, and so on, may produce improvedperformance.

In one controller optimization example, assume that the plant is minimumphase, (e.g., its poles and zeros are in the left half plane), that theplant transfer function is given, that the ω_(c) parameterizedcontrollers are known and available in form of Table I, that a transientprofile is defined according to the transient response specifications,and that a simulator 800 of closed-loop control system as shown in FIG.8 is available. The simulator 800 includes a transient profile generator810, a digital controller 820, and plant or system G_(p)(s) 830. It isto be appreciated that the closed loop control system simulator 800 canbe, for example, hardware, software or a combination of both. In oneexample, the simulator incorporates limiting factors including, but notlimited to, sensor and quantization noises, sampling disturbances,actuator limits, and the like.

With these assumptions, one example design method then includes,determining frequency and gain scales, ω_(p) and k from the given planttransfer function. The method also includes, based on the designspecification, determining the type of controller required from, forexample, Table I. The method also includes selecting the G_(c)(s, ω_(c))corresponding to the scaled plant in the form of Table I. The methodalso includes scaling the controller to

${\frac{1}{k}{G_{c}\left( {\frac{s}{\omega_{p}},\omega_{c}} \right)}},$digitizing G_(c)(s/ω_(p), ω_(c))/k and implementing the controller inthe simulator. The method may also include setting an initial value ofω_(c) based on the bandwidth requirement from the transient response andincreasing ω_(c) while performing tests on the simulator, until eitherone of the following is observed:

-   -   a. Control signal becomes too noisy and/or too uneven, or    -   b. Indication of instability (oscillatory behavior)

Consider an example motion control test bed for which the mathematicalmodel of the motion system isÿ=(−1.41{dot over (y)}+23.2T _(d))+23.2u  (38)where y is the output position, u is the control voltage sent to thepower amplifier that drives the motor, and T_(d) is the torquedisturbance. An example design objective for the example system could berotating the load one revolution in one second with no overshoot. Thus,the physical characteristics of the example control problem are:

1) |u|<3.5 volt,

2) sampling rate=1 kHz,

3) sensor noise is 0.1% white noise,

4) torque disturbance up to 10% of the maximum torque,

5) smooth control signal.

The plant transfer function is

${{G_{p}(s)} = \frac{k}{\frac{s}{\omega_{p}}\left( {\frac{s}{\omega_{p}} + 1} \right)}},{k = {{11.67\mspace{14mu}{and}\mspace{14mu}\omega_{p}} = {1.41.}}}$Now consider the corresponding UGUB plant

${{\overset{\_}{G}}_{p}(s)} = {\frac{1}{s\left( {s + 1} \right)}.}$A PD design ofu=k _(p)(r−y)+k _(d)(−{dot over (y)})with

-   -   k_(p)=ω_(c) ² and k_(d)=2ω_(c)−1        makes the closed-loop transfer function

${G_{c1}(s)} = {\frac{\omega_{c}^{2}}{\left( {s + \omega_{c}} \right)^{2}}.}$Considering the plant gain scale of k and the frequency scale of ω_(p),the PD gains are then scaled as

$k_{p} = {\frac{\omega_{c}^{2}}{k} = {{{.086}\mspace{11mu}\omega_{c}^{2}\mspace{14mu}{and}\mspace{14mu} k_{d}} = {\frac{{2\;\omega_{c}} - 1}{k\;\omega_{p}} = {{.061}{\left( {{2\;\omega_{c}} - 1} \right).}}}}}$To avoid noise corruptions of the control signal, an approximatedifferentiator

$\frac{s}{\left( {\frac{s}{{10\;\omega_{c}}\;} + 1} \right)^{2}}$is used where the corner frequency 10ω_(c) is selected so that thedifferentiator approximation does not introduce problematic phase delaysat the crossover frequency. Using a conventional root locus method, theone second settling time would require a closed-loop bandwidth of 4rad/sec. The example single parameter design and tuning methodsdescribed herein facilitate determining that an ω_(c) of 20 rad/secyields optimal performance under the given conditions. A comparison ofthe two designs is shown in FIG. 9. Note that a step disturbance of 1volt is added at t=3 seconds to test disturbance rejection. Finally, atrapezoidal transient profile is used in place of the step command. Theresults are shown in FIG. 10.Parameterization of State Feedback and State Observer Gains

As described in the Background section, the State Feedback (SF)controlleru=r+K{circumflex over (x)}  (4)is based on the state space model of the plant:{dot over (x)}(t)=Ax(t)+Bu(t), y(t)=Cx(t)+Du(t)  (5)When the state x is not accessible, a state observer (SO):{circumflex over ({dot over (x)}=A{circumflex over (x)}+Bu+L(y−ŷ)  (6)is often used to find its estimate, {circumflex over (x)}. Here r is thesetpoint for the output to follow. The state feedback gain K and theobserver gain L are determined from the equations:eig(A+BK)=λ_(c)(s) and eig(A+LC)=λ_(o)(s)where λ_(c)(s) and λ_(o)(s) are polynomials of s that are chosen by thedesigner. Usually the K and L have many parameters and are hard to tune.

The parameterization of state feedback and state observer gains areachieved by making

-   -   λ_(c)(s)=(s+ω_(c))^(n) and λ_(o)(s)=(s+ω_(o))^(n)        where ω_(c) and ω_(o) are bandwidth of the state feedback system        and the state observer, respectively, and n is the order of the        system. This simplifies tuning since parameters in K and L are        now functions of ω_(c) and ω_(o), respectively.        Parameterization of Linear Active Disturbance Rejection        Controller (LADRC) for a Second Order Plant

Some controllers are associated with observers. Conventionally, secondorder systems with controllers and observers may have a large number(e.g., 15) tunable features in each of the controller and observer.Thus, while a design method like the Hann method is conceptually viable,its practical implementation is difficult because of tuning issues. As aconsequence of the scaling and parameterization described herein,observer based systems can be constructed and tuned using twoparameters, observer bandwidth (ω₀) and controller bandwidth (ω_(c)).

State observers provide information on the internal states of plants.State observers also function as noise filters. A state observer designprinciple concerns how fast the observer should track the states, (e.g.,what should its bandwidth be). The closed-loop observer, or thecorrection term L(y−ŷ) in particular, accommodates unknown initialstates, uncertainties in parameters, and disturbances. Whether anobserver can meet the control requirements is largely dependent on howfast the observer can track the states and, in case of ESO, thedisturbance f(t,x₁,x₂,w). Generally speaking, faster observers arepreferred. Common limiting factors in observer design include, but arenot limited to dependency on the state space model of the plant, sensornoise, and fixed sampling rate.

Dependency on the state space model can limit an application tosituations where a model is available. It also makes the observersensitive to the inaccuracies of the model and the plant dynamicchanges. The sensor noise level is hardware dependent, but it isreasonable to assume it is a white noise with the peak value 0.1% to 1%of the output. The observer bandwidth can be selected so that there isno significant oscillation in its states due to noises. A state observeris a closed-loop system by itself and the sampling rate has similareffects on the state observer performance as it does on feedbackcontrol. Thus, an example model independent state observer system isdescribed.

Observers are typically based on mathematical models. Example systemsand methods described herein can employ a “model independent” observeras illustrated in FIG. 18. For example a plant 1820 may have acontroller 1810 and an observer 1830. The controller 1810 may beimplemented as a computer component and thus may be programmaticallytunable. Similarly, the observer 1830 may be implemented as a computercomponent and thus may have scaleable parameters that can be scaledprogrammatically. Furthermore, using analogous scaling andparameterizing as described herein, the parameters of the observer 1830can be reduced to ω_(o). Therefore, overall optimizing of the system1800 reduces to tuning ω_(c) and ω_(o).

Consider a simple example for controlling a second order plantÿ=−a{dot over (y)}−by+w+bu  (39)where y and u are output and input, respectively, and w is an inputdisturbance. Here both parameters, a and b, are unknown, although thereis some knowledge of b, (e.g., b₀≈b, derived from the initialacceleration of y in step response). Rewrite (39) asÿ=−a{dot over (y)}−by+w+(b−b ₀)u+b _(o) u=f+b ₀u  (40)where f=−a{dot over (y)}−by+w+(b−b_(o))u. Here f is referred to as thegeneralized disturbance, or disturbance, because it represents both theunknown internal dynamics, −a{dot over (y)}−by+(b−b_(o))u and theexternal disturbance w(t).

If an estimate of f, {circumflex over (f)} can be obtained, then thecontrol law

$u = \frac{{- \hat{f}} + u_{0}}{b_{0}}$reduces the plant to ÿ=(f−{circumflex over (f)})+u_(o) which is aunit-gain double integrator control problem with a disturbance(f−{circumflex over (f)}).

Thus, rewrite the plant in (40) in state space form as

$\begin{matrix}\left\{ \begin{matrix}{{{\overset{.}{x}}_{1} = x_{2}}\mspace{65mu}} \\{{\overset{.}{x}}_{2} = {x_{3} + {b_{0}u}}} \\{{{\overset{.}{x}}_{3} = h}\mspace{76mu}} \\{{y = x_{1}}\mspace{59mu}}\end{matrix} \right. & (41)\end{matrix}$with x₃=f added as an augmented state, and h={dot over (f)} is seen asan unknown disturbance. Now f can be estimated using a state observerbased on the state space model

$\begin{matrix}{{\overset{.}{x} = {{A\; x} + {B\; u} + {E\; h}}}{y = {C\; z}}{where}\text{}{{A = \begin{bmatrix}0 & 1 & 0 \\0 & 0 & 1 \\0 & 0 & 0\end{bmatrix}},{B = \begin{bmatrix}0 \\b_{0} \\0\end{bmatrix}},{C = \left\lbrack {1\mspace{14mu} 0\mspace{14mu} 0} \right\rbrack},{E = \begin{bmatrix}0 \\0 \\1\end{bmatrix}}}} & (42)\end{matrix}$Now the state space observer, denoted as the linear extended stateobserver (LESO), of (42) can be constructed asż=Az+Bu+L(y−ŷ)ŷ=Cz  (43)

which can be reconstructed in software, for example, and L is theobserver gain vector, which can be obtained using various methods knownin the art like pole placement,L=[β ₁β₂β₃]^(T)  (44)where [ ]^(T) denotes transpose. With the given state observer, thecontrol law can be given as:

$\begin{matrix}{u = \frac{{- z_{3}} + u_{0}}{b_{0}}} & (45)\end{matrix}$Ignoring the inaccuracy of the observer,ÿ=(f−z ₃)+u ₀ ≈u _(o)  (46)which is an unit gain double integrator that can be implemented with aPD controlleru ₀ =k _(p)(r−z ₁)−k _(d) z ₂  (47)where r is the setpoint. This results in a pure second order closed-looptransfer function of

$\begin{matrix}{G_{cl} = \frac{1}{s^{2} + {k_{d}s} + k_{p}}} & (48)\end{matrix}$Thus, the gains can be selected ask _(d)=2ξω_(c) and k_(p)=ω_(c) ²  (49)where ω_(c) and ζ are the desired closed loop natural frequency anddamping ratio. ζ can be chosen to avoid oscillations. Note that−k_(d)z₂, instead of k_(d)({dot over (r)}−z₂), is used to avoiddifferentiating the setpoint and to make the closed-loop transferfunction a pure second order one without a zero.

This example illustrates that disturbance observer based PD controlachieves zero steady state error without using the integral part of aPID controller. The example also illustrates that the design is modelindependent in that the design relies on the approximate value of b in(39). The example also illustrates that the combined effects of theunknown disturbance and the internal dynamics are treated as ageneralized disturbance. By augmenting the observer to include an extrastate, it is actively estimated and canceled out, thereby achievingactive disturbance rejection. This LESO based control scheme is referredto as linear active disturbance rejection control (LADRC) because thedisturbance, both internal and external, represented by f, is activelyestimated and eliminated.

The stability of controllers can also be examined. Lete_(i)=x_(i)−z_(i), i=1, 2, 3. Combine equation (43) and (44) andsubtract the combination from (42). Thus, the error equation can bewritten:

$\begin{matrix}{{\overset{.}{e} = {{A_{e}e} + {Eh}}}{where}{A_{e} = {{A - {LC}} = \begin{bmatrix}{- \beta_{1}} & 1 & 0 \\{- \beta_{2}} & 0 & 1 \\{- \beta_{3}} & 0 & 0\end{bmatrix}}}} & (50)\end{matrix}$and E is defined in (42). The LESO is bounded input, bounded output(BIBO) stable if the roots of the characteristic polynomial of A_(e)λ(s)=s ³+β₁ s ²+β₂ s+β ₃  (51)are in the left half plane (LHP) and h is bounded. This separationprinciple also applies to LADRC.

The LADRC design from (43) to (46) yields a BIBO stable closed-loopsystem if the observer in (43) and (44) and the feedback control law(46) for the double integrator are stable, respectively. This is shownby combing equations (45) and (47) into a state feedback form ofu=(1/b₀)[−k_(p)−k_(d)−1]z=Fz, where F=(1/b₀)[−k_(p)−k_(d)−1]. Thus, theclosed-loop system can be represented by the state-space equation of:

$\begin{matrix}{\begin{bmatrix}\overset{.}{x} \\\overset{.}{z}\end{bmatrix} = {{\begin{bmatrix}A & {\overset{\_}{B}F} \\{LC} & {A - {LC} + {\overset{\_}{B}F}}\end{bmatrix}\begin{bmatrix}x \\z\end{bmatrix}} + {\begin{bmatrix}\left\lbrack \overset{\_}{B} \right. & \left. E \right\rbrack \\\overset{\_}{\left\lbrack B \right.} & \left. 0 \right\rbrack\end{bmatrix}\begin{bmatrix}r \\h\end{bmatrix}}}} & (52)\end{matrix}$where B=B/b₀, and which is BIBO stable if its eigenvalues are in theLHP. By applying row and column operations, the closed-loop eigenvalues

${{eig}\left( \begin{bmatrix}A & {\overset{\_}{B}F} \\{LC} & {A - {LC} + {\overset{\_}{B}F}}\end{bmatrix} \right)} = {{eig}\left( \begin{bmatrix}{A + {\overset{\_}{B}F}} & {\overset{\_}{B}F} \\0 & {A - {LC}}\end{bmatrix} \right)}$ $\begin{matrix}{= {{{eig}\left( {A + {\overset{\_}{B}F}} \right)} \Cup {{eig}\left( {A - {LC}} \right)}}} \\{= {\left\{ {{{roots}\mspace{14mu}{of}\mspace{14mu} s^{2}} + {k_{d}s} + k_{p}} \right\} \Cup \left\{ \mspace{11mu}{{{roots}\mspace{20mu}{of}\mspace{14mu} s^{3}} + {\beta_{1}s^{2}} + {\beta_{2}s} + \beta_{3}} \right\}}}\end{matrix}$

Since r is the bounded reference signal, a nontrivial condition on theplant is that h={dot over (f)} is bounded. In other words, thedisturbance f must be differentiable.

Observer Bandwidth Parameterization

ω_(o) parameterization refers to parameterizing the ESO on observerbandwidth ω_(o). Consider a plant (42) that has three poles at theorigin. The related observer will be less sensitive to noises if theobserver gains in (44) are small for a given ω_(o). But observer gainsare proportional to the distance for the plant poles to those of theobserver. Thus the three observer poles should be placed at −ω_(o), orequivalently,λ(s)=s ³+β₁ s ²+β₂ s+β ₃=(s+ω _(o))³  (53)That isβ₁=3ω_(o), β₂=3ω_(o) ², β₃=ω_(o) ³  (54)

It is to be appreciated by one of ordinary skill in the art thatequations (53) and (54) are extendable to nth order ESO. Similarly, theparameterization method can be extended to the Luenberger Observer forarbitrary A, B, and C matrices, by obtaining {Ā, B, C} as observablecanonical form of {A,B,C}, determining the observer gain, L, so that thepoles of the observer are at −ω_(o) and using the inverse statetransformation to obtain the observer gain, L, for {A,B,C}. Theparameters in L are functions of ω_(o). One example procedure for ω_(o)optimization based design is now described.

Given tolerable noise thresholds in the observer states, increase ω_(o)until at least one of the thresholds is about to be reached or theobserver states become oscillatory due to sampling delay. In general,the faster the ESO, the faster the disturbance is observed and cancelledby the control law.

A relationship between ω_(o) and ω_(c) can be examined. One examplerelationship isω_(o)≈3␣5ω_(c)  (55)

Equation (55) applies to a state feedback control system where ω_(c) isdetermined based on transient response requirements like the settlingtime specification. Using a transient profile instead of a step commandfacilitates more aggressive control design. In this example there aretwo bandwidths to consider, the actual control loop bandwidth ω_(c) andthe equivalent bandwidth of the transient profile, ω _(c). Part of thedesign procedure concerns selecting which of the two to use in (55).Since the observer is evaluated on how closely it tracks the states andω _(c) is more indicative than ω_(c) on how fast the plant states move,ω _(c) is the better choice although it is to be appreciated that eithercan be employed. Furthermore, taking other design issues like thesampling delay into consideration, a more appropriate minimum ω_(o) isfound through simulation and experimentation asω_(o)≈5␣10 ω _(c)  (56)

An example for optimizing LADRC is now presented. One example LADRCdesign and optimization method includes designing a parameterized LESOand feedback control law where ω_(o) and ω_(c) are the designparameters. The method also includes designing a transient profile withthe equivalent bandwidth of ω _(c) and selecting an ω_(o) from (56). Themethod then includes setting ω_(c)=ω_(o) and simulating and/or testingthe LADRC in a simulator. The method also includes incrementallyincreasing ω_(c) and ω_(o) by the same amount until the noise levelsand/or oscillations in the control signal and output exceed thetolerance. The method also includes incrementally increasing ordecreasing ω_(c) and ω_(o) individually, if necessary, to maketrade-offs between different design considerations like the maximumerror during the transient period, the disturbance attenuation, and themagnitude and smoothness of the controller.

In one example, the simulation and/or testing may not yield satisfactoryresults if the transient design specification described by ω _(c) isuntenable due to noise and/or sampling limitations. In this case,control goals can be lowered by reducing ω _(c) and therefore ω_(c) andω_(o). It will be appreciated by one skilled in the art that thisapproach can be extended to Luenberg state observer based state feedbackdesign.

By way of illustration, reconsider the control problem exampleassociated with equations (32), but apply the LADRC in (43) to (48).Note that b=23.2 for this problem, but to make the design realistic,assume the designer's estimate of b is b₀=40. Now rewrite the plantdifferential equation (38) asÿ(−1.41{dot over (y)}+23.2T _(d))+(23.2−40)u+40u=f+40uThe LESO is

$\overset{.}{z} = {{\begin{bmatrix}{{- 3}\;\omega_{o}} & 1 & 0 \\{{- 3}\;\omega_{o}^{2}} & 0 & 1 \\{- \omega_{o}^{3}} & 0 & 0\end{bmatrix}z} + {\begin{bmatrix}0 & {3\;\omega_{o}} \\40 & {3\;\omega_{o}^{2}} \\0 & \omega_{o}^{3}\end{bmatrix}\begin{bmatrix}u \\y\end{bmatrix}}}$andz₁→y,z₂→{dot over (y)}, andz ₃ →f=−1.41{dot over (y)}+23.2T _(d)+(23.2−40)u, as t→∞The control law is defined as

$u = {{\frac{u_{0} - z_{3}}{40}\mspace{14mu}{and}\mspace{14mu} u_{0}} = {{k_{p}\left( {r - z_{1}} \right)} - {k_{d}z_{2}}}}$with

-   -   k_(d)=2ξω_(c), ξ=1, and k_(p)=ω_(c) ²        where ω_(c) is the sole design parameter to be tuned. A        trapezoidal transient profile is used with a settling time of        one second, or ω _(c)=4. From (56), ω_(o) is selected to be 40        rad/sec. The LADRC facilitates design where a detailed        mathematical model is not required, where zero steady state        error is achieved without using the integrator term in PID,        where there is better command following during the transient        stage and where the controller is robust. This performance is        achieved by using a disturbance observer. Example performance is        illustrated in FIG. 12.        Parameterization of LADRC for nth Order Plant

It will be appreciated by one skilled in the art that observer baseddesign and tuning techniques can be scaled to plants of arbitraryorders. For a general nth order plant with unknown dynamics and externaldisturbances,y ^((n)) =f(t,y,{dot over (y)}, . . . , y ^((n−1)) ,u,{dot over (u)}, .. . u ^((n−1)) ,w)+bu  (57)the observer can be similarly derived, starting from the state spaceequation

$\begin{matrix}\left\{ \begin{matrix}{{\overset{.}{x}}_{1} = x_{2}} \\{{\overset{.}{x}}_{2} = x_{3}} \\\cdots \\{{\overset{.}{x}}_{n} = {x_{n + 1} + {b_{0}u}}} \\{{\overset{.}{x}}_{n + 1} = h} \\{y = x_{1}}\end{matrix} \right. & (58)\end{matrix}$with x_(n+1)=f added as an augmented state, and h=f mostly unknown. Theobserver of (43) in its linear form with the observer gainL=[β ₁β₂ . . . β_(n+1)]  (59)has the form

$\begin{matrix}\left\{ \begin{matrix}{{\overset{.}{z}}_{1} = {z_{2} - {\beta_{1}\left( {z_{1} - {y(t)}} \right)}}} \\{{\overset{.}{z}}_{2} = {z_{3} - {\beta_{2}\left( {z_{1} - {y(t)}} \right)}}} \\\cdots \\{{\overset{.}{z}}_{n} = {z_{n + 1} - {\beta_{n}\left( {z_{1} - {y(t)}} \right)} + {b_{0}u}}} \\{{\overset{.}{z}}_{n + 1} = {- {\beta_{n + 1}\left( {z_{1} - {y(t)}} \right)}}}\end{matrix} \right. & (60)\end{matrix}$With the gains properly selected, the observer will track the states andyieldz₁(t)→y(t),z₂(t)→{dot over (y)}(t), . . . ,Z_(n)(t)→y^((n−1))(t)z_(n+1)(t)→f(t,y,{dot over (y)}, . . . ,y^((n−1),u,{dot over (u)}, . . . u) ^((n−1),w))  (61)The control law can also be similarly designed as in (45) and (47), with

$\begin{matrix}{u = \frac{z_{n + 1} + u_{0}}{b_{0}}} & (62)\end{matrix}$which reduces the plant to approximately a unit gain cascaded integratorplanty ^((n))=(f−z _(n+1))+u ₀ ≈u ₀  (63)andu ₀ =k _(p)(r−z ₁)−k _(d) ₁ z ₂ − . . . −k _(d) _(n−1) z _(n)  (64)where the gains are selected so that the closed-loop characteristicpolynomial has n poles at −ω_(c),s ^(n) +k _(d) _(n−1) s ^(n−1) + . . . +k _(d) ₁ s+k _(p)=(s+ω_(c))^(n)  (65)ω_(c) is the closed-loop bandwidth to be optimized in tuning. The ω_(o)optimization can similarly be applied usings ^(n)+β₁ s ^(n−1)+ . . . +β_(n−1) s+β _(n)=(s+ω _(o))^(n)  (66)

The following example method can be employed to identify a plant orderand b₀. Given a “black box” plant with input u and output y, the order,n, and b₀ can be estimated by allowing the plant to discharge energystored internally so that it has a zero initial condition, (e.g.,y(0)={dot over (y)}(0)= . . . y^((n−1))(0)=0) and then assuming f(0)=0.The method includes applying a set of input signals and determining theinitial slope of the response: {dot over (y)}(0⁺),ÿ(0⁺), . . . . Themethod also includes determining the slope y^((i))(0⁺) that isproportional to u(0) under various tests, (e.g., y^((i))(0⁺)=ku(0)).Then the method includes setting n=i+1 and b₀=k.

Auto-Tuning Based on the New Scaling, Parameterization and OptimizationTechniques

Auto-tuning concerns a “press button function” in digital controlequipment that automatically selects control parameters. Auto-tuning isconventionally realized using an algorithm to calculate the PIDparameters based on the step response characteristics like overshoot andsettling time. Auto-tuning has application in, for example, the start upprocedure of closed-loop control (e.g., commissioning an assembly linein a factory). Auto-tuning can benefit from scaling andparameterization.

In some applications, dynamic changes in the plant during operations areso severe that controller parameters are varied from one operating pointto another. Conventionally, gain-scheduling is employed to handle thesesituations. In gain-scheduling, the controller gains are predeterminedfor different operating points and switched during operations.Additionally, and/or alternatively, self-tuning that actively adjustscontrol parameters based on real time data identifying dynamic plantchanges is employed.

Common goals of these techniques are to make the controller parameterdetermination automatic, given the plant response to a certain inputexcitation, say a step function and to maintain a consistent controllerperformance over a wide range of operations, (e.g. making the controllerrobust).

Example systems, methods and so on described herein concerning scalingand parameterization facilitate auto-scaling model based controllers.When a transfer function model of a plant is available, the controllercan be designed using either pole placement or loop shaping techniques.Thus, example scaling techniques described herein facilitate automatingcontroller design and tuning for problems including, but not limited to,motion control, where plants are similar, differing only in de gain andthe bandwidth, and adjusting controller parameters to maintain highcontrol performance as the bandwidth and the gain of the plant changeduring the operation.

In the examples, the plant transfer functions can be represented as G_(p)(s)=kG_(p)(s/ω_(p)), where G_(p)(s) is given and known as the“mother” plant and k and ω_(p) are obtained from the plant response ortransfer function. Assuming the design criteria are similar in nature,differing only in terms of the loop gain bandwidth, ω_(c), thecontroller for similar plants can be automatically obtained by scalingthe given controller, G_(c)(s,ω_(c)), for G_(p)(s). This is achieved bycombining the controller scaling, defined in equation (26), andω_(c)-parameterization to obtain the controller for G_(p)(s)=kG_(p)(s/ω_(p)) asG _(c)(s,ω _(c))=G _(c)(s/ω _(p), ω_(c))/k  (67)

There are three parameters in (67) that are subject to tuning. The firsttwo parameters, k and ω_(p), represent plant changes or variations thatare determined. The third parameter, ω_(c), is tuned to maximizeperformance of the control system subject to practical constraints.

An example method for auto-tuning is now described. The auto-tuningmethod includes examining a plant G_(p)(s) and the nominal controllerG_(c)(s,ω_(c)). Given the plant G_(p)(s) and the nominal controllerG_(c)(s,ω_(c)), the method includes performing off-line tests todetermine k and ω_(p) for the plant. The method also includes usingequation (67) to determine a new controller for the plant, G_(p)(s)=kG_(p)(s/ω_(p)), obtained in the previous act. The method alsoincludes optimizing ω_(c) for the new plant.

An example method for adaptive self-tuning is now described. Theadaptive self-tuning procedure includes examining a plant G_(p)(s)=kG_(p)(s/ω_(p)), where k and ω_(p) are subject to change duringplant operation. Given the plant G _(p)(s)=kG_(p)(s/ω_(p)), the methodincludes performing real time parameter estimation to determine k andω_(p) as they change. The method also includes determining when theperformance of the control system is degraded beyond a pre-determined,configurable threshold and updating the controller using (67). Themethod also includes selectively decreasing ω_(c) if the plant dynamicsdeviate significantly from the model kG_(p)(s/ω_(p)), which causesperformance and stability problems. The method also includes selectivelyincreasing ω_(c) subject to ω_(c)-optimization constraints if the plantmodel can be updated to reflect the changes of the plant beyond k andω_(p).

The LADRC technique does not require the mathematical model of theplant. Instead, it employs a rough estimate of the single parameter b inthe differential equation of the plant (57). This estimation is denotedas b₀ and is the sole plant parameter in LADRC. As the dynamics of theplant changes, so does b. Thus, b₀ can be estimated by rewriting (57) asy ^((n)) =f(t)+bu  (69)and assuming the zero initial condition, (e.g., y^((i))(0)=0, i=1, 2, .. . n−1 and f(0)=0). Then b₀≈b can be estimated by usingb _(o) =y ^((n))(0⁺)/u(0)  (70)where u(0) is the initial value of the input. It is to be appreciatedthat this method can be applied to both open loop and closed-loopconfigurations. For the auto-tuning purposes, the test can be performedoff-line and a step input, u(t)=constant can be applied. The LADRC doesnot require b₀ to be highly accurate because the difference, b−b₀, istreated as one of the sources of the disturbance estimated by LESO andcancelled by control law.

The b₀ obtained from the off-line estimation of b described above can beadapted for auto-tuning LADRC. An auto-tuning method includes,performing off-line tests to determine the order of the plant and b₀,selecting the order and the b₀ parameter of the LADRC using the resultsof the off-line tests, and performing a computerized auto-optimization.

Using the controller scaling, parameterization and optimizationtechniques presented herein, an example computer implemented method 1300as shown in FIG. 13 can be employed to facilitate automaticallydesigning and optimizing the automatic controls (ADOAC) for variousapplications. The applications include, but are not limited to, motioncontrol, thermal control, pH control, aeronautics, avionics,astronautics, servo control, and so on.

The method 1300, at 1310, accepts inputs including, but not limited to,information concerning hardware and software limitations like theactuator saturation limit, noise tolerance, sampling rate limit, noiselevels from sensors, quantization, finite word length, and the like. Themethod also accepts input design requirements like settling time,overshoot, accuracy, disturbance attenuation, and so on. Furthermore,the method also accepts as input the preferred control law form like,PID form, model based controller in a transfer function form, and modelindependent LADRC form. In one example, the method can indicate if thecontrol law should be provided in a difference equation form. At 1320, adetermination is made concerning whether a model is available. If amodel is available, then at 1330 the model is accepted either intransfer function, differential equations, or state space form. If amodel is not available, then the method may accept step response data at1340. Information on significant dynamics that is not modeled, such asthe resonant modes, can also be accepted.

Once the method has received information input, the method can checkdesign feasibility by evaluating the specification against thelimitations. For example, in order to see whether transientspecifications are achievable given the limitations on the actuator,various transient profiles can be used to determine maximum values ofthe derivatives of the output base on which the maximum control signalcan be estimated. Thus, at 1350, a determination is made concerningwhether the design is feasible. In one example, if the design is notfeasible, processing can conclude. Otherwise, processing can proceed to1360.

If the input information passes the feasibility test, then at 1360, themethod 1300 can determine an ω^(c) parameterized solution in one or moreformats. In one example, the ω_(c) solution can then be simulated at1370 to facilitate optimizing the solution.

In one example, to assist an engineer or other user, the ADOAC methodprovides parameterized solutions of different kind, order, and/or forms,as references. The references can then be ranked separately according tosimplicity, command following quality, disturbance rejection, and so onto facilitate comparison.

FIG. 14 illustrates a computer 1400 that includes a processor 1402, amemory 1404, a disk 1406, input/output ports 1410, and a networkinterface 1412 operably connected by a bus 1408. Executable componentsof the systems described herein may be located on a computer likecomputer 1400. Similarly, computer executable methods described hereinmay be performed on a computer like computer 1400. It is to beappreciated that other computers may also be employed with the systemsand methods described herein. The processor 1402 can be a variety ofvarious processors including dual microprocessor and othermulti-processor architectures. The memory 1404 can include volatilememory and/or non-volatile memory. The non-volatile memory can include,but is not limited to, read only memory (ROM), programmable read onlymemory (PROM), electrically programmable read only memory (EPROM),electrically erasable programmable read only memory (EEPROM), and thelike. Volatile memory can include, for example, random access memory(RAM), synchronous RAM (SRAM), dynamic RAM (DRAM), synchronous DRAM(SDRAM), double data rate SDRAM (DDR SDRAM), and direct RAM bus RAM(DRRAM). The disk 1406 can include, but is not limited to, devices likea magnetic disk drive, a floppy disk drive, a tape drive, a Zip drive, aflash memory card, and/or a memory stick. Furthermore, the disk 1406 caninclude optical drives like, compact disk ROM (CD-ROM), a CD recordabledrive (CD-R drive), a CD rewriteable drive (CD-RW drive) and/or adigital versatile ROM drive (DVD ROM). The memory 1404 can storeprocesses 1414 and/or data 1416, for example. The disk 1406 and/ormemory 1404 can store an operating system that controls and allocatesresources of the computer 1400.

The bus 1408 can be a single internal bus interconnect architectureand/or other bus architectures. The bus 1408 can be of a variety oftypes including, but not limited to, a memory bus or memory controller,a peripheral bus or external bus, and/or a local bus. The local bus canbe of varieties including, but not limited to, an industrial standardarchitecture (ISA) bus, a microchannel architecture (MSA) bus, anextended ISA (EISA) bus, a peripheral component interconnect (PCI) bus,a universal serial (USB) bus, and a small computer systems interface(SCSI) bus.

The computer 1400 interacts with input/output devices 1418 viainput/output ports 1410. Input/output devices 1418 can include, but arenot limited to, a keyboard, a microphone, a pointing and selectiondevice, cameras, video cards, displays, and the like. The input/outputports 1410 can include but are not limited to, serial ports, parallelports, and USB ports.

The computer 1400 can operate in a network environment and thus isconnected to a network 1420 by a network interface 1412. Through thenetwork 1420, the computer 1400 may be logically connected to a remotecomputer 1422. The network 1420 can include, but is not limited to,local area networks (LAN), wide area networks (WAN), and other networks.The network interface 1412 can connect to local area networktechnologies including, but not limited to, fiber distributed datainterface (FDDI), copper distributed data interface (CDDI),ethernet/IEEE 802.3, token ring/IEEE 802.5, and the like. Similarly, thenetwork interface 1412 can connect to wide area network technologiesincluding, but not limited to, point to point links, and circuitswitching networks like integrated services digital networks (ISDN),packet switching networks, and digital subscriber lines (DSL).

Referring now to FIG. 15, information can be transmitted between variouscomputer components associated with controller scaling andparameterization described herein via a data packet 1500. An exemplarydata packet 1500 is shown. The data packet 1500 includes a header field1510 that includes information such as the length and type of packet. Asource identifier 1520 follows the header field 1510 and includes, forexample, an address of the computer component from which the packet 1500originated. Following the source identifier 1520, the packet 1500includes a destination identifier 1530 that holds, for example, anaddress of the computer component to which the packet 1500 is ultimatelydestined. Source and destination identifiers can be, for example,globally unique identifiers (guids), URLS (uniform resource locators),path names, and the like. The data field 1540 in the packet 1500includes various information intended for the receiving computercomponent. The data packet 1500 ends with an error detecting and/orcorrecting field 1550 whereby a computer component can determine if ithas properly received the packet 1500. While six fields are illustratedin the data packet 1500, it is to be appreciated that a greater and/orlesser number of fields can be present in data packets.

FIG. 16 is a schematic illustration of sub-fields 1600 within the datafield 1540 (FIG. 15). The sub-fields 1600 discussed are merely exemplaryand it is to be appreciated that a greater and/or lesser number ofsub-fields could be employed with various types of data germane tocontroller scaling and parameterization. The sub-fields 1600 include afield 1610 that stores, for example, information concerning thefrequency of a known controller and a second field 1620 that stores adesired frequency for a desired controller that will be scaled from theknown controller. The sub-fields 1600 may also include a field 1630 thatstores a frequency scaling data computed from the known frequency andthe desired frequency.

Referring now to FIG. 17, an application programming interface (API)1700 is illustrated providing access to a system 1710 for controllerscaling and/or parameterization. The API 1700 can be employed, forexample, by programmers 1720 and/or processes 1730 to gain access toprocessing performed by the system 1710. For example, a programmer 1720can write a program to access the system 1710 (e.g., to invoke itsoperation, to monitor its operation, to access its functionality) wherewriting a program is facilitated by the presence of the API 1700. Thus,rather than the programmer 1720 having to understand the internals ofthe system 1710, the programmer's task is simplified by merely having tolearn the interface to the system 1710. This facilitates encapsulatingthe functionality of the system 1710 while exposing that functionality.Similarly, the API 1700 can be employed to provide data values to thesystem 1710 and/or retrieve data values from the system 1710.

For example, a process 1730 that retrieves plant information from a datastore can provide the plant information to the system 1710 and/or theprogrammers 1720 via the API 1700 by, for example, using a call providedin the API 1700. Thus, in one example of the API 1700, a set ofapplication program interfaces can be stored on a computer-readablemedium. The interfaces can be executed by a computer component to gainaccess to a system for controller scaling and parameterization.Interfaces can include, but are not limited to, a first interface 1740that facilitates communicating controller information associated withPID production, a second interface 1750 that facilitates communicatingplant information associated with PID production, and a third interface1760 that facilitates communicating frequency scaling informationgenerated from the plant information and the controller information.

The systems, methods, and objects described herein may be stored, forexample, on a computer readable media. Media can include, but are notlimited to, an ASIC, a CD, a DVD, a RAM, a ROM, a PROM, a disk, acarrier wave, a memory stick, and the like. Thus, an example computerreadable medium can store computer executable instructions for one ormore of the claimed methods.

What has been described above includes several examples. It is, ofcourse, not possible to describe every conceivable combination ofcomponents or methodologies for purposes of describing the systems,methods, computer readable media and so on employed in scaling andparameterizing controllers. However, one of ordinary skill in the artmay recognize that further combinations and permutations are possible.Accordingly, this application is intended to embrace alterations,modifications, and variations that fall within the scope of the appendedclaims. Furthermore, the preceding description is not meant to limit thescope of the invention. Rather, the scope of the invention is to bedetermined only by the appended claims and their equivalents.

While the systems, methods and so on herein have been illustrated bydescribing examples, and while the examples have been described inconsiderable detail, it is not the intention of the applicants torestrict or in any way limit the scope of the appended claims to suchdetail. Additional advantages and modifications will be readily apparentto those skilled in the art. Therefore, the invention, in its broaderaspects, is not limited to the specific details, the representativeapparatus, and illustrative examples shown and described. Accordingly,departures may be made from such details without departing from thespirit or scope of the applicant's general inventive concept.

1. A method for processing controller parameters, comprising: selectinga controller to parameterize; identifying two or more gains for thecontroller; mathematically relating the two or more gains to a singlecontroller tuning parameter to yield a parameterized controller, whereinthe mathematically relating includes generating, for the two or moregains, respective two or more functions, wherein an only variable of therespective two or more functions is the single controller tuningparameter, wherein the single controller tuning parameter is a controlsystem bandwidth ω_(c) of the controller, and wherein the control systembandwidth ω_(c) defines a common location of N poles of an Nth orderclosed-loop system comprising the parameterized controller and a plant,where N is an integer; and relating the parameterized controller withthe single controller tuning parameter to the plant to yield theclosed-loop system.
 2. The method of claim 1, wherein the selecting thecontroller comprises selecting a controller that is aproportional/integral/derivative (PID) controller having parameters thatinclude one or more of proportional gain k_(p), integral gain k_(i), orderivative gain k_(d), wherein the one or more of the proportional gaink_(p), the integral gain k_(i)or the derivative gain k_(d) are definedas functions of only the control system bandwidth ω_(c).
 3. The methodof claim 1, further comprising relating a model of the controller G_(c)to a model of the plant G_(p) in terms of the single controller tuningparameter ω_(c,), a Laplace Transform variable s, and a coefficient ξ according to: G_(p)(s) $\frac{1}{s + 1}$ $\frac{1}{s}$$\frac{1}{s^{2} + {2\xi\; s} + 1}$ $\frac{1}{s\left( {s + 1} \right)}$$\frac{1}{s^{2}}$ G_(c)(s, ω_(c))$\frac{\omega_{c}\left( {s + 1} \right)}{s}$ ω_(c)$\omega_{c}^{2}\frac{s^{2} + {2\xi\; s} + 1}{s\left( {s + {2\omega_{c}}} \right)}$$\frac{\omega_{c}^{2}\left( {s + 1} \right)}{s + {2\omega_{c}}}$$\frac{\omega_{c}^{2}s}{s + {2\omega_{c}}}.$


4. The method of claim 1, further comprising obtaining a model of thecontroller G_(c) via a loop shaping design having a form of equation(36): $\begin{matrix}{{G_{c}(s)} = {\left( \frac{s + \omega_{1}}{s} \right)^{m}\frac{1}{\frac{s}{\omega_{c}} + 1}\frac{1}{\left( {\frac{s}{\omega_{2}} + 1} \right)^{n}}{G_{p}^{- 1}(s)}}} & (36)\end{matrix}$ wherein n is such that$\frac{1}{\frac{s}{\omega_{c}} + 1}\frac{1}{\left( {\frac{s}{\omega_{c}} + 1} \right)^{n}}{G_{p}^{- 1}(s)}$is proper and G_(p) is a model of the plant, ω_(c) is the control systembandwidth of the controller, ω₁ is a lower constraint, ω₂ is an upperconstraint, and n and m are both integers.
 5. The method of claim 1,wherein the selecting the controller comprises selecting a controllerthat is a state feedback (SF) controller.
 6. The method of claim 5,wherein the closed-loop system includes a state observer (SO) having anobserver gain L.
 7. The method of claim 1, wherein the closed-loopsystem includes a linear extended state observer (LESO).
 8. The methodof claim 1, wherein the controller is a linear active disturbancerejection controller (LADRC).
 9. The method of claim 1, whereinselecting the controller comprises selecting a transfer function based(TFB) controller.
 10. The method of claim 6, further comprisingparameterizing the observer gain L to be a function of a single tuningparameter ω_(o), wherein ω_(o) is an observer bandwidth.
 11. The methodof claim 7, further comprising parameterizing the LESO to be a functionof a single tuning parameter ω_(o), wherein ω_(o) is an observerbandwidth, and wherein M observer poles are placed at −ω_(o) for an Mthorder plant, where M is an integer.
 12. A method for processingcontroller parameters, comprising: selecting a controller for control ofan Nth order plant, wherein the controller is defined by two or morecontroller coefficients; placing N poles of the controller at a commonlocation defined by a controller bandwidth ω_(c) , of the controller,where N is an integer; parameterizing the controller to generate aparameterized controller, wherein the parameterizing includes definingthe two or more controller coefficients using different algebraicfunctions expressed in terms of only the controller bandwidth ω_(c);relating the parameterized controller to the Nth order plant to yield aclosed-loop system comprising the parameterized controller and the Nthorder plant; and configuring the parameterized controller for control ofthe Nth order plant by selecting a value for the controller bandwidth.13. The method of claim 1, wherein the N poles of an Nth orderclosed-loop system are located at −ω_(c).
 14. The method of claim 13,further comprising tuning the controller for control of the plant byoptimizing the control system bandwidth ω_(c).
 15. The method of claim5, wherein the state feedback controller has a form of equation (4):u=r+K{circumflex over (x)}  (4) where u is a control input, r is areference, K is a state feedback matrix, and {circumflex over (x)} is anestimated state vector.
 16. The method of claim 6, wherein the stateobserver has a form of equation (6):{circumflex over (x)}=A{circumflex over (x)}+Bu+L(y−ŷ)  (6) where{circumflex over (x)} is a derivative of the estimated state vector, Ais a state matrix, B is an input matrix, L is an observer gain, y is anoutput, and ŷ is an observed output.
 17. The method of claim 7, whereinthe LESO has a form of equation (43) for a 2^(nd) order plant and a formof equation (60) for an n^(th) order plant:ż=Az+Bu+L(y−ŷ)  (43) $\begin{matrix}\left\{ \begin{matrix}{{\overset{.}{z}}_{1} = {z_{2} - {\beta_{1}\left( {z_{1} - {y(t)}} \right)}}} \\{{\overset{.}{z}}_{2} = {z_{3} - {\beta_{2}\left( {z_{1} - {y(t)}} \right)}}} \\\cdots \\{{\overset{.}{z}}_{n} = {z_{n + 1} - {\beta_{n}\left( {z_{1} - {y(t)}} \right)} + {b_{0}u}}} \\{{\overset{.}{z}}_{n + 1} = {- {\beta_{n + 1}\left( {z_{1} - {y(t)}} \right)}}}\end{matrix} \right. & (60)\end{matrix}$ where z is an observer state, ż is a derivative of theobserver state, A is a state matrix, B is an input matrix, L is anobserver gain, y is an output, ŷ is an observed output, and β_(i), i=1,2, ...n+1, are coefficients of the observer gain.
 18. The method ofclaim 8, wherein the LADRC has a form of equations (43), (44), (45),(47), (49) and (54) for a 2^(nd) order plant and a form of equations(60), (62), (64), (65), and (66) for an n^(th) order plant:ż=Az+Bu+L(y−ŷ)ŷ=Cz  (43)L=[β ₁β₂β₃]^(T)  (44) $\begin{matrix}{u = \frac{{- z_{3}} + u_{0}}{b_{0}}} & (45)\end{matrix}$u ₀ =k _(p)(r−z ₁)−k _(d) z ₂  (47)k_(d)=2ξω_(c) and k_(p)=ω_(c) ²  (49)β₁=3ω_(o), β₂=3ω_(o) ², β₃=ω_(o) ³  (54) $\begin{matrix}\left\{ \begin{matrix}{{\overset{.}{z}}_{1} = {z_{2} - {\beta_{1}\left( {z_{1} - {y(t)}} \right)}}} \\{{\overset{.}{z}}_{2} = {z_{3} - {\beta_{2}\left( {z_{1} - {y(t)}} \right)}}} \\\cdots \\{{\overset{.}{z}}_{n} = {z_{n + 1} - {\beta_{n}\left( {z_{1} - {y(t)}} \right)} + {b_{0}u}}} \\{{\overset{.}{z}}_{n + 1} = {- {\beta_{n + 1}\left( {z_{1} - {y(t)}} \right)}}}\end{matrix} \right. & (60)\end{matrix}$ $\begin{matrix}{u = \frac{z_{n + 1} + u_{0}}{b_{0}}} & (62)\end{matrix}$u ₀ =k _(p)(r−z ₁)−k _(d) ₁ z ₂ − . . . −k _(d) _(n−1) z _(n)  (64)s ^(n) +k _(d) _(n−1) s ^(n−1) + . . . +k _(d) ₁ s+k _(p)=(s+ω_(c))^(n)  (65)s ^(n)+β₁ s ^(n−1)+ . . . +β_(n−1) s+β _(n)=(s+ω _(o))^(n)  (66) where zis an observer state, {circumflex over (z)} is a derivative of theobserver state, A is a state matrix, B is an input matrix, L is anobserver gain, y is an output, ŷ is an observed output, β_(n) arecoefficients of the observer gain, u is a control input, {circumflexover (x)} is an estimated state vector, k are gains, ω_(c) is thecontrol system bandwidth, and ω_(o) is an observer bandwidth.
 19. Themethod of claim 1, wherein the selecting the controller comprises:selecting a known controller for a second plant that is analogous to thefirst plant; identifying one or more scalable parameters of the knowncontroller, the one or more scalable parameters comprising at least oneof a frequency or a gain; and scaling the one or more scalableparameters of the known controller to yield the controller.
 20. Themethod of claim 19, wherein the scaling comprises scaling the one ormore scalable parameters based on at least one of an identifiedfrequency scale or an identified gain scale of the second plant.
 21. Themethod of claim 12, wherein the placing the N poles comprises placingthe N poles at ω_(c).